3 THE MODULUS FUNCTION ON A GROUP 255 0,1, 2,..., iQ? — 1), the residues mod p of the numbers z* are all distinct). Deduce that in this case y~p. Hence, using the multiplicative property of norms of quater- nions, deduce that every positive integer is the sum of at most four squares (Lagrange^s theorem). 3. (a) Let A be Lebesgue measure on R. Let/be a real-valued function i>0 on R which is A-integrable, bounded and of compact support. Put y— sup/(0- For each weR, r eR let U/w) denote the set of t e R such that/(/) ^ w, and put vf(w) = A*(U/O)). Show that, for all a > 1, /•+«> -y f*(t)dt= J -oo Jo -1 dw. (b) Let g be another function satisfying the same conditions as/, and put 3 = sup g(t). reR Let h be the function on R2 defined by h(u, v) =/O) + g(v) if f(u)g(v)^Q> and h(u,v) = Q otherwise; also let k(t)= sup h(u,v), so that k is positive, A-integrable, bounded and compactly supported. Show that, for all a > 1, r+°o / 1 r + oo i r + co \ fc«« dtZ(Y+Sr[—\ f'(f) dt+j-\ g'(t) dt . J-oo \y J-oo Oj_oo / (Observe that if 0 < w ^ 1 we have \Jk(yw + Bw) ^ Uf(yw) -h U/(3H>), and use (a) above and Problem 4(d) of Section 14.1). (c) Let An be Lebesgue measure on R", and let A, B be two An-integrable subsets of R". Show that ((A«)*(A + B))1'" £ (An(A))1/n ~h (AnCB))1/" (Brunn-Minkowski inequality). (Reduce to the case where A and B are compact. Then use induction on n, Problem 4(d) of Section 14.1, the theorem of Lebesgue-Fubini, the inequality established in (b) above, and Holder's inequality.) 4. Let p be a prime number. The normalized Haar measure p, on the compact group Zp of p-adic integers (Section 12.9, Problem 4) is such that the measure of any closed ball of radius p"~k is equal to p~k. (Show that Zp is the union of pk closed balls of radius /?""*, no two of which intersect.) 3. THE MODULUS FUNCTION ON A GROUP; THE MODULUS OF AN AUTOMORPHISM Let G be a locally compact group, ju a left Haar measure on G. For all s, t e G we have directly from the definitions (14.1.2), and therefore &(s)n is also a left-invariant positive measure on G. Hence there is a unique real number AG(X) > 0 (also written A(s)) such that $(S)JLI = AG(X)ju (14.1.5), and this number is clearly, by (/(O dt)/t, is independent of the choice of interval